Neutrino physics, like leptonic CP violation, is an interesting topic^[1] in the current research of particle physics. Among other things, it might be the final place where experiments of particle physics will give definite results in the near future. The results will check various theoretical models about the fermion masses of the Standard Model (SM).

We proposed that supersymmetry (SUSY)^[2] can be the theory underlying the fermion masses in Refs. [3–5]. The basic idea is the following. It assumes a flavor symmetry. The flavor symmetry is broken after the sneutrinos obtain nonvanishing vacuum expectation values (VEVs). (In this way, SUSY is motivated.) These VEVs result in a nonvanishing neutrino mass. The empirical smallness of neutrino masses needs very large SM super partner masses to be understood which are about 10¹² GeV. Thus, our SUSY is of high scale breaking.^[6–8]

A further natural assumption is that the flavor symmetry breaks softly. Namely the soft SUSY breaking masses of the sfermions do not obey the flavor symmetry either. The theoretical reason is that the soft masses are due to the supergravity effect which generically breaks any global symmetry. Soft breaking of the flavor symmetry implies that the lepton number violation due to sneutrino VEVs is explicit instead of being spontaneous. Therefore there is no any massless Nambu-Goldstone boson related to nonvanishing sneutrino VEVs. Actually the large masses of the model make the low energy effective theory just the SM via Higgs mass fine tuning, except for that we have an understanding of the hierarchical pattern of the charged lepton masses, or that of the SM Yukawa coupling constants.

Let us briefly review the model in a simple way. The SM is SUSY generalized. The flavor symmetry is Z₃ cyclic among the three generation SU(2)_L lepton doublets L₁, L₂ and L₃. The other fields are trivial under Z₃. The Z₃ invariant combinations are and with α and β denoting the SU(2)_L indices. In terms of the following redefined lepton superfields, , , , the above Z₃ invariant combinations are and , respectively. The superpotential is then
where H_u and H_d are the two Higgs doublets, the right-handed lepton singlet is defined as the one which couples to L_τ, and is that orthogonal to and with a coupling to . y_τ, and are coupling constants. (Note that considering the mixing between and H_d gives the same form of the above superpotential.^[4]) It is seen that the electron is massless, because is always absent in the Lagrangian. This is true whenever SUSY is conserved, the nonvanishing electron mass is due to SUSY breaking (together with electroweak gauge symmetry and flavor symmetry breaking via loops). Note that all the coupling constants in our superpotential are assumed to be natural values, say typically ∼0.01-1, and the mass parameter is taken to be large GeV. The SM fermion mass hierarchy is due to symmetries and their breaking.

In addition, a heavy vector-like SU(2)_L triplet field with hypercharge 2(−2) needs to be introduced so as to make the Higgs mass realistic.^[5–6] This triplet field also contributes to neutrino masses. In terms of the redefined fields, the flavor symmetric superpotential relevant to the triplet T and fields is
with M_T the mass GeV. The braces denote that the two doublets form an SU(2)_L triplet representation.

The soft SUSY breaking terms in the Lagrangian are in general form, which also break the flavor symmetry.^[3–5] All the mass parameters of the model are taken to be about (10¹²–10¹³) GeV. The spontaneous gauge symmetry breaking of the SM occurs. Through fine tuning, the right electroweak vacuum is obtained. By including contribution due to the triplet field, this model can give reasonable neutrino spectrum and the mixing pattern, and predicted the right order of θ₁₃.^[4,5] (The quark sector was considered in Ref. [4].)

Roughly speaking about the electroweak symmetry breaking. There are five scalar doublets, the mass parameters are all large GeV. Eigenvalues of their mass-squared matrix are generically large. However, one of these values can be exceptional, because it is a difference between two large parameters. It is this difference that makes the fine-tuning possible. Whence the difference is tuned to be about −(100 GeV)², correct electroweak symmetry breaking occurs. The corresponding eigenstate field is one superposition of the five doublets. It is the only light scalar doublet, and is just the SM Higgs field from the point of view of the low energy effective field theory. The SM Higgs gets a VEV is equivalent to that the original two Higgses and sleptons get their VEVs.^[4–5]

In this paper, we will carefully consider CP violation of the lepton sector, and completely analyze the neutrino masses and mixing. In general, the coupling constants are complex, however, because of the flavor symmetry, many of them can be made real via field phase rotation. In the superpotential Eq. (1) for charged leptons, all the couplings can be adjusted to be real. On the other hand, in the superpotential Eq. (2) for neutrino masses, the couplings cannot be all taken real, as can be seen in the following way. The mass parameters and M_T are taken real, thus H_u and H_d always have opposite phases, and so do T and . is real via rotating the phase of , is real via rotating H_u (or ), is real via , real via rotating, and real via . In such a phase convention, only , and can be complex. The term will contribute to the neutrino masses, which was omitted in our previous analysis.^[5]

In the soft SUSY breaking terms, the mass parameters and coupling constants are generally complex, and there is no enough freedom to rotate all of the phases away.

The scalar potential relevant to the electroweak symmetry breaking is
where g and are SM gauge coupling constants. h_u and h_d denote the scalar components of H_u and H_d, respectively, and ʼs left-handed sleptons. , , and , are soft squared masses.

In considering CP violation of the scalar potential, the essential point lies in the soft bilinear terms where the mass parameters are complex. Field redefinition of h_d and may remove phases of and respectively, however, the phases of and off-diagonal terms of are still there. This means that after the electroweak symmetry breaking, Higgs and sneutrino VEVs are complex in general. (Previously we took all the VEVs real.) In the analysis, we still have the freedom to choose the VEV of Higgs field h_u to be real, and VEVs of the Higgs and the sneutrino fields are denoted as (v_u, , , , ) where the phases have been explicitly written down. These VEVs enter the lepton mass matrices and thus contribute to CP violation in the leptonic mixing.

The sneutrino VEVs result in a nonvanishing neutrino mass,
where , is the Zino mass, which is the typical superpartner mass, and the phase of Zino mass term, , is explicitly written. This is due to gauge interactions, it is natural realization of the type-I seesaw mechanism^[9] where the role of right-handed neutrinos is replaced by the Zino. In addition, the superpotential (2) contributes the following neutrino masses,^[5]
where the phase of coupling has been explicitly written. This part of neutrino mass generation is realization of the type-II seesaw mechanism.^[10]

The full neutrino mass matrix is
Note this is the full neutrino mass matrix of the model. It is due to tree level contribution of lepton number violation. The loop level contribution due to R-parity violation is negligible,^[4] because the sparticles in the loops are very heavy.

The physics analysis including is different from our previous one.^[5] We observe that it is natural to take that is numerically dominant over , then there appears a degeneracy between the first two neutrinos. This roughly fits the neutrino spectrum obtained from neutrino oscillation experiments. This degeneracy is perturbed by which also contributes neutrino mixing. Furthermore, it is interesting to note that inclusion of in certain cases does not really increase difficulty in the analysis because is diagonal.

We rewrite by adjusting the diagonal part to be proportional to identity matrix,
where
and
where , , and . Generally, is complex, the phases make further analytical calculation^[11] difficult. For illustration and an easy analysis, and without losing generality about CP violation, we simply take and in the following. Then, up to an overall factor, is a real symmetric matrix and can be diagonalized by an orthogonal matrix. It just needs diagonalizing , because is essentially a unit matrix, which does not affect this diagonalization. By further assuming that , which is reasonable because which does not violate the Z₃ flavor symmetry, it is found that is diagonalized by,
with eigenvalues
In fact, diagonalizes ,
Noticing that the diagonalized matrix is still complex, we further write that
the neutrino masses in our model are
with the phases

It is clear that and are almost degenerate with a mass . Their mass splitting is about . and have the same order of magnitude by definition, and we take . According to neutrino oscillation experiments,^[12] and , this model typically gives that
Naturally the phases in above formulae are . This makes us to take all the cosines to be for simplicity in estimating the neutrino masses. And is numerically fixed by choosing and .

Finally, we obtain the unitary matrix U_ν which diagonalizes ,

with P being the pure phase matrix appearing in Eq. (13).

From Eq. (1), the charged lepton mass matrix is obtained. Considering the sneutrino and Higgs VEVs are complex, it is
Here the electron mass is neglected. In this model, the electron mass would be a loop contribution of SUSY breaking terms, which also break the flavor symmetry and the electroweak symmetry.^[3,4] The analytical expressions of and are obtained as
The mass matrix M^l basically fixes the mixing due to charged leptons with a precision of . It is standard to find the unitary matrix U_l which diagonalizes ,
It can be expressed as
where

The lepton mixing matrix is . It is obtained that mixing is
The mixing is

The mixing is
Experimental data for best values of these mixings are , , and .^[12] Obviously, taking , is in agreement with data. The value of is taken to be larger and still in the natural range, . Choosing , it is easy to get .

For , there are two terms in Eq. (25), neglecting the first term for simplicity, this mixing would be maximal if , namely . Of course, a smaller is more natural. That is to say, the atmospheric neutrino angle tends to be that . However if is larger, is also achievable. Therefore this model just slightly favors the atmospheric neutrino angle to be in the first octant.

The important CP violation in neutrino oscillations is given through the invariant parameter J,^[13]

is expected to be large, namely . This agrees with current preliminary experimental results.^[15]

The effective Majorana mass in the neutrinoless double beta decay is
In this work, it is
In the above formula, the term has a Majorana phase dependence, which is negligibly small anyway.

Like gauge theories which are used to describe the elementary particle interactions, SUSY is used for fermion masses. Our model is the minimal SUSY SM with a vector-like triplet field extension, but SUSY breaks at a high scale and the R-parity (lepton number) is not required. The sneutrino VEVs result in a neutrino mass, which is suppressed by the Zino mass. This is a nice realization of the type-I seesaw mechanism which, even does not need to introduce any right-handed neutrino. The triplet field is originally for the realistic Higgs mass. However, it also contributes to neutrino masses through a type-II seesaw mechanism. The Zino related seesaw mechanism results in only one massive neutrino. By including the triplet contribution, the neutrino masses can be realistic. Compared to our previous studies,^[4–5] a more natural pattern for neutrino masses is obtained.

To be numerically natural, let us return back to the original superpotential in the beginning. The couplings are assumed to be taken natural values. The field VEVs are mainly fixed by the soft parameters in the Lagrangian, in addition to those in the superpotential. To fit the lepton spectrum and mixing, we take , , , , and . Note does not break the flavor symmetry, it is natural that its value is more close to v_d. And the large ratio is for explaining the top quark mass.^[4] When , and are in the same order, the correct neutrino spectrum is obtained. In terms of parameters in the superpotential, we have . , and λʼs .

It is necessary to check the reliability of our approximation in estimating the neutrino masses. That approximation about the phases can be good when the quantities appear in the mass formulae are hierarchical, say if . As it has been seen that this is indeed the case for . In (Eq. (14)), , , and are of the same order. This allows us to look at an extreme case where the phase is π. In this case, there is a possibility of inverted neutrino mass hierarchy, namely a very small . But this is achieved through a large cancellation between and . Although this is possible, it is unnatural.

The physics of neutrinos in this work is quite different from that in Refs. [4–5]. This is mainly due to the triplet. In Ref. [4], we introduced a singlet, the neutrino mass matrix was that with only the 33 matrix element nonvanishing. And in Ref. [5], the triplet replaced the singlet for the Higgs mass in the beginning, however, in the neutrino mass analysis, we took to be zero which essentially was the same as that for the singlet case. Taking to be zero was actually unreasonable because our principle is to treat all the basic couplings close to . As a result, in Refs. [4,5], there was always one massless neutrino. That led to that the Majorana mass is about 10⁻³ eV. In addition, in Ref. [5] it was wrong to say CP violation is small in the lepton sector.

In summary, in the model of high scale SUSY for understanding the fermion mass hierarchies, we have studied CP violation in the lepton sector, and other aspects of neutrino physics in detail. In the analysis, the phases of the Higgs and sneutrino VEVs, and contribution of the term in superpotential (2), have been included. This analysis is more complete than previous consideration. The neutrino mass matrix, and the charged lepton one, are fixed by the model. Its specific feature is the triplet contribution, the approximate degeneracy of neutrinos ν₁ and ν₂ can be naturally explained.

This model could not predict exact values of the fermion masses because of the flavor symmetry breaking as well as SUSY breaking. However, the principle we follow is that all the coupling constants should be in the natural parameter range which is about (0.01–1). Taking triplet contribution dominant, and inputting relevant experimental data on leptons, we obtain that (i) , , . This normal ordering neutrino spectrum is to be checked in JUNO experiment.^[14] (ii) CP violation in neutrino oscillation most probably is large. There have been some experimental hint on this.^[15] CP violation in neutrino oscillations is a great study task experimentally.^[16] (iii) The effective Majorana neutrino mass in the neutrinoless double beta decay is about 0.02 eV, it is within the detection ability of future measurements.^[17] (iv) θ₂₃ is only slightly favored being in the first octant. (v) The electron neutrino mass to be measured in β decays is about 0.02 eV. This is, however, still one order of magnitude lower than the future limit of direct measurements.^[18] (vi) The sum of three neutrino masses is close to . If the standard cosmology is correct, astrophysics measurements on the cosmic microwave background have constrained this sum to be .^[19] It is interesting to note that a recent analysis showed the sum is about ∼0.11 eV.^[20] Most of the above predictions are close to their experimental limits, therefore, this model will soon be checked experimentally.

We would like to thank Gui-Jun Ding and Zhen-Hua Zhao for very helpful discussions.